The Bull Lurcher is a highly focused, very smart and easily excited dog whose love for the hunt is practically second to none. Still, these are dogs that love their master and tend to have a very strong loyalty to just one person. They love all people, but they often single out the one person on whom they rely for food, etc.
They do have a remarkably strong prey drive that is more and more exploited by the kind of people who get a thrill from dog-fighting. Because off this, this sighthound mix is too often believed to be inherently aggressive. They will never not want to chase small furry animals, but they can be trained to resist as well as properly restrained. Their prey drive can also be redirected toward toys or handled with food rewards. Understanding his prey drive can also help. For instance, if he starts zig-zagging, then he's on a scent and soon to chase.
These are some extremely strong, tireless, and active dogs. They need a lot of exercise. It's best if they have a very well-fenced outdoor area in which to run long and free. Two walks a day otherwise may not be enough. They are keen to learn, to leap, and to use their great strength when playing. You should have lots of durable agility and strength toys. You might even want to build an exercise course for them.